Transversals of Latin squares and covering radius of sets of permutations

We consider the symmetric group S n as a metric space with the Hamming metric. The covering radius cr ( S ) of a set of permutations S ⊂ S n is the smallest r such that S n is covered by the balls of radius r centred at the elements of S . For given n and s , let f ( n , s ) denote the cardinality of the smallest set S of permutations with cr ( S ) ⩽ n − s . lue of f ( n , 2 ) is the subject of a conjecture by Kézdy and Snevily that implies two famous conjectures by Ryser and Brualdi on transversals in Latin squares. We show that f ( n , 2 ) ⩽ n + O ( log n ) for all n and that f ( n , 2 ) ⩽ n + 2 whenever n = 3 m for m > 1 . We also construct, for each odd m ⩾ 3 , a Latin square of order 3 m with two rows that each contain 2 m − 2 transversal-free entries. This gives an infinite family of Latin squares with odd order n and at most n / 3 + O ( 1 ) disjoint transversals. The previous strongest upper bound for such a family was n / 2 + O ( 1 ) . y, we show that f ( 5 , 3 ) = 15 and record a proof by Blackburn that cr ( AGL ( 1 , q ) ) = q − 3 when q is odd.